Question 870229
.
A river flows with uniform velocity v. A person in a motorboat travels 1 km upstream, 
at which time a log is seen floating by. 
The person continues to travel upstream for one more hour at the same speed 
and then returns downstream to the starting point, where the same log is seen again. 
Find the velocity of the river. 
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            The solution to this problem by @mananth in the archive is  INCORRECT.


            He incorrectly interpreted the problem and produced  WRONG  solution.


            Ignore his post.


            Below find my correct solution.



<pre>
Let u be the speed of the boat in still water (in km/h).

and let v be the speed of the river ( == the speed of the current ).


(1)  First, person travels 1 km upstream.



(2)  At that time, he sees the log floating by.  Let call this position "the meeting point".



(3)  The person continue to travel upsteam for 1 more hour.


     During this hour, the person travels (u-v) km upstream.
     
     During this hour, the log travels v kilometers downstream with the current.



(4)  Then the person turns back, travels downstream and reaches the starting point at the same time as the log comes to this point.


         +----------------------------------------------+
         |   Let's consider kinematic, starting from    |
         |   the "meeting point" to the ending moment.  |
         +----------------------------------------------+


    During this time, the log travel 1 km with the current speed.



    The motorboat travels 1 hour u-v kilometers upstream, and then travels  ((u-v)+1)) kilometers DOWNSTREAM with the speed of (u+v) km/h.



    So, the log's travel time from the meeting point to the end is  {{{1/v}}}  hours;

        the motorboat's travel time from the meeting point to the end is  {{{(u-v+1)/(u+v) + 1}}}  hours.



    The two travel times are the same, giving this "time equation"

        {{{1/v}}} = {{{(u-v+1)/(u+v)}}} + 1   hours.



    OK. The setup is done.  Now our task is to solve the equation.

        For it, multiply both sides by v*(u+v).  You will get


            u + v = v*(u-v+1) + v*(u+v)

            u + v = vu - v^2 + v + vu + v^2

            u     = vu           + vu

            u     =  2vu

     Now divide both sides by u (which is not a zero, so we can divide safely)  and get

            1     = 2v

            v = {{{1/2}}}.


The problem is just solved ( by a miraculous way (!) ).


<U>ANSWER</U>.  The speed of the river (== the speed of the current) is  {{{1/2}}}  km/h.
</pre>

Solved, and carefully explained.



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It is one of the classic, &nbsp;advanced and the most &nbsp;"delicious" &nbsp;Travel & Distance &nbsp;problems.



The key point in the solution is to get the setup equation, &nbsp;which I called &nbsp;"the time equation".



As soon as you get it, &nbsp;the rest is just a technique.


It is just a technique, but one miraculous moment still takes place (!)


This miraculous moment is the fact that the problem allows us to get a &nbsp;UNIQUE &nbsp;SOLUTION &nbsp;to equation in two unknowns &nbsp;(&nbsp;!&nbsp;)



Thanks to the mother-nature and to persons who &nbsp;invented/composed  &nbsp;this miraculous problem many years ago &nbsp;(&nbsp;!&nbsp;)